1. ## Continuity

Hey.

Could someone help me with the following?

A function f on R to R satisfies for all x,y
abs(f(x)-f(y)) less than or equal to k*abs(x-y) where k is a constant greater than zero.

First, prove that f is continuous. Also, use this to show that given an epsilon greater than 0, there is at least one delta greater than zero that "measures" the continuity for this epsilon at all points.

Thanks a lot. I have no idea what I'm doing.

2. Originally Posted by JackStolerman
A function f on R to R satisfies for all x,y
abs(f(x)-f(y)) less than or equal to k*abs(x-y) where k is a constant greater than zero.
First, prove that f is continuous.
Of course, this is a Lipschitz condition. It not only implies continuity, but uniform continuity!
It works for any $K > 0$.
To see that: given $\varepsilon > 0$ choose $\delta = \frac{\varepsilon }{K}$.

Originally Posted by JackStolerman
show that given an epsilon greater than 0, there is at least one delta greater than zero that "measures" the continuity for this epsilon at all points.
I am not sure what that means. K is sometimes called the Lipschitz Constant.